This short note is an analysis of the foundation result of and a theory application of Ross McPherson's1 astute observation that the pure number ratio RPE of the electromagnetic potential energy between a proton and an electron to the gravitational potential energy between a proton and an electron is very well approximated by the quantum frequency νp of the proton multiplied by P=1016. Such an identification using P, a power of ten integer, or with some other close by numerical multiplier, if firmly established, could very well lead to a quantum-electromagnetic theory of gravity as suggested by Ross McPherson. Here, I shall work with an exact relation between the proton's rest mass electromagnetic potential energy frequency νp = mpc2/h and the proton's internal gravitational potential energy frequency which can be obtained by suitably defining the gravitational potential energy frequency νG,p associated with the proton's rest mass. The application objective here is to show that Newton's Gravitational coupling constant G 4,5,6,7 can be classified as a memberof the set of quantum coupling constants that the author 2,3 has suggested are involved in the quantum context. Historically RPE is one of the three very large pure numbers that featured in the works of (Eddington, Dirac, Jordon, Dick, Hayakawa, Carter)7 and others so that certainly it has been very extensively studied but even so it is still within the category of mysterious numbers another relevant one being 137. This is the application aspect I wish to examine in this brief article.
The dimensionless ratio, RPE, can be defined as
| RPE = α h' c/(G mp me), | 1 |
where α is the fine structure constant, h' is the symbol I shall use here for Planck's constant h divided by 2 π. That is h' = hbar = h/(2 π), c is the speed of light, G is Newton's gravitation constant, mp is the rest mass of the proton, me is the rest mass of the electron. RPE is essentially the ratio of the electromagnetic coupling constant α and the dimensioned gravitational coupling constant G.
The quantity,
| GEQ = G mpme /(h'c) | 2 |
can be regarded as a differently valued dimensionless version of G or the electromagnetic-quantum value for gravitational coupling. Thus the dimensionless ratio or pure number RPE can be represented as,
| RPE = α/GEQ, | 3 |
and RPE can be seen as a conversion factor that converts, by multiplication, gravitational potential energy to electromagnetic-quantum potential energy. Generally in quantum theory associated with an energy E there is a frequency given by ν = E/h. Thus the zero dimensioned multiplication factor RPE will similarly convert gravitational frequency to electromagnetic-frequency. The constituent physical quantities involved in the definition of RPE are α, h', c, G, mp and me all of which have approximate measured physical values that can be found in CODATA's6 set of recommended values. Using these values the consequent approximate largest value of this ratio is found to be
| RPE = α/GEQ = 2.27251×1039, | 4 |
| RPE = α/GEQ = 2.26571×1039, | 5 |
for the two cases of smallest G and greatest G from the CODATA value
| G = 6.673(10)×10 -11. | 6 |
McPherson observation is that the quantum frequency for the proton's rest mass has the value
| νp = mpc2/h = 2.268732×10 23, | 7 |
Thus this value, if multiplied by P = 1016, lies between the lower and upper values of RPE and so P νp could be the true physical value of RPE. However, we can take a more cautious approach and note that we can multiply the value of νp by a factor P' = 1016ς and provided ς is suitably close to unity the result would still lie between the lower and upper values of RPE. I prefer to take this more general view of the situation because it is obviously mathematically correct and does not interfer with the physical meaning of the observation or raise any questions of accuracy. McPherson's identification is just the case when ς = 1. The value for ς = 1 also gives McPherson value for G as in equation (8) but more generally for values ς ≈ 1 and suitably limited a good value for G can be calculated from,
| G = α h' c 10 -16/(ς mp me νp), | 8 |
which will be close to the measured CODATA value for G and it seems to me to open the door for the construction of an electromagnetic theory of gravitation. The use of the paramter ς represents taking a more relaxed view of CODATA's recommended values while giving more credence to the range of the experimental results in which the true values should certainly occur. However, there are two points that need to be made about this identification. Unless the quantity ς is taken to have the units form of time the identification will be physically meaningless. The other point is that ς can also be taken to have values close to unity and an electromagnetic theory of gravity could still follow in the McPherson pattern but with slightly different numerical values involved. I shall now complete this short article by using equation (1) to define the gravitation potential energy of the proton and show how that can be used to express RPE in terms of my set of quantum coupling constants. From the discussion above and to avoid any dimensional mismatch we can define the gravitational frequency, νG,p, associated with the protons rest mass as
| νG,p = 10 -16/ς , | 9 |
in general and without prejudice to whatever near to unity value ς may have. Importantly ς will be taken to have the units form time. Thus the dimensionless ratio can be written as,
| RPE = α/GEQ = νp/νG,p. | 10 |
The first form gives a clear indication how the dimensionless set of quantum coupling constants should be associated with this ratio. The fine structure constant α is given by the function α(137,29) and clearly the dimensionless gravitational constant GEQ should be associated with a member of the set of quantum coupling constants. Suppose this coupling constant is taken to be the member of the set α(N1,N2) where the two integers N1,N2 are to be determined. The dimensionless ratio now becomes
| RPE = α(137,29)/α(N1,N2). | 11 |
It follows from equation (1) that
| G = α(N1,N2) h' c/(mp me). | 12 |
because the fine structure constant α = α(137,29). Equation (12) firmly places Newton's gravitation coupling constant amongst all the other quantum coupling constants in the set {α(n1,n2)}3. However, The values of the integer parameters N1,N2 remain undetermined. We can however, put into equation (12) some good value of G and solve for N1 call it NG and use the measured values for G, h', mp, me and c to obtain the value of NG. I shall take the value of N2 = 1 because in essence this is a refinement parameter much outweighed by the very large N1 value. Following the calculation through, the gravitational quantum number NG from the current value of G is found to be
| NG ≈ 3.1095 × 1041. | 13 |
This 42 digit integer has been calculated to only 5 significant integers because the experimental input data is very limited. In fact, the numerical value of G alone is only known with certainty to 2 significant figures. From the point of view of this article the integer NG is to be put in the same category as the quantum number 137 not obtainable from existing theory but nevertheless an extremely important but possible mysterious number. It does however, have 42 digits so to get it exactly will undoubtedly be extremely difficult or even an impossible task.
| 1. | Ross McPherson, Electrifying Gravity (Sept, 2004) |
| 2. | J. G. Gilson, Speculations in Science and Technology 17 (3), 201 (1994) |
| 3. | J. G. Gilson, Physics Essays vol. 9, No. 2, (1996) |
| 4. | A. Sommerfeld, Ann. Phys. 51, 1 (1916) |
| 5. | W. Rindler, Relativity, Special, General and Cosmological, Oxford University Press (2001) |
| 6. | P. J. Mohr, B. N. Taylor, Journal of Chemical and Physical Reference Data Vol. 28 No 6, pp 1713-1852 (1999) |
| 7. | C. Misner, K. Thorn, J. A. Wheeler, Gravitation, Freeman,(1973), Page 1216 |